An a posteriori error estimate for a linear-nonlinear transmission problem in plane elastostatics

We consider the coupling of dual-mixed finite element and boundary element methods to solve a linear-nonlinear transmission problem in plane hyperelasticity with mixed boundary conditions. Besides the displacement and the stress tensor, we introduce the strain tensor as an additional unknown, which yields a two-fold saddle point operator equation as the corresponding variational formulation. We derive a reliable a posteriori error estimate that depends on the solution of local Dirichlet problems and on residual terms on the transmission and Neumann boundaries, which are given in a negative order Sobolev norm. Our approach does not need the exact Galerkin solution, but any reasonable approximation of it. In addition, the analysis does not depend on special finite element or boundary element subspaces. However, for certain specific subspaces we are able to provide two fully local a posteriori error estimates, in which the residual terms are bounded by weighted local L ²-norms. Further, one of the error estimates does not require the explicit solution of the local problems. Received: November 2000 / Revised version: December 2001 This research was partially supported by Fondecyt-Chile through research projects 1980122 and 2000124, and the FONDAP Program in Applied Mathematics, and by the Dirección de Investigación of the Universidad de Concepción through the Advanced Research Groups Program.     

A transient finite element solution method for the Navier-Stokes equations

This paper is concerned with the discrete finite element formulation and numerical solution of transient incompressible viscous flow in terms of the primitive variables. A restricted variational principle is introduced as equivalent to the momentum equations and the Poisson equation for pressure. The latter is introduced to replace the continuity equation, and thus the incompressibility condition is realized only asymptotically; i.e. through the iterative process. An incomplete cubic interpolation function is used for both the velocities and pressure within a triangular finite element. The discrete equations are integrated in time with backward finite differences. We illustrate the similarity between the (ψ,ζ) finite difference method and the ($\text{u}\text{,p}$) finite element method by calculations on the driven square cavity problem.     

Superconvergence of Finite Element Approximations for the Fractional Diffusion-Wave Equation

In this paper, the error estimates of fully discrete finite element approximation for the time fractional diffusion-wave equation are discussed. Based on the standard Galerkin finite element method approach for the spatial discretization and the L1 formula for the approximation of the time fractional derivative, the fully discrete scheme for solving the constant coefficient fractional diffusion-wave equation is obtained and the superconvergence estimate is proposed and analyzed. Further, a fully discrete finite element scheme is presented for solving the variable coefficient fractional diffusion-wave equation and the corresponding error estimates are also established. Finally, numerical experiments are included to support the theoretical results.     

Maximum principle and uniform convergence for the finite element method

The solution of the Dirichlet boundary value problem over a polyhedral domain Ω ? Rⁿ, n ≥ 2, associated with a second-order elliptic operator, is approximated by the simplest finite element method, where the trial functions are piecewise linear. When the discrete problem satisfies a maximum principle, it is shown that the approximate solution u_h converges uniformly to the exact solution u if u ? W_1,p (Ω), with p > n, and that $\text{∥u?u}{}_{\mathrm{h}}\text{∥}{}_{\mathrm{L}}\text{∞(Ω) = O(h)}$ if u ? W²,p(Ω), with 2p > n. In the case of the model problem ?Δu+au = f in Ω, u = u_o on δΩ, with a ? 0, a simple geometrical condition is given which insures the validity of the maximum principle for the discrete problem.     

Solution errors in finite element analysis

The development of the finite element method so far indicates that it is a discretization technique especially suited for positive definite, self-adjoint, elliptic systems, or systems with such components. The application of the method leads to the discretized equations in the form of $\text{u}\text{?}\text{=}\text{f(u)}$, where u lists the response of the discretized system at n preselected points called nodes. Instead of explicit expressions, vector function f and its Jacobian f,_u are available only numerically for a numerically given u. The solution of $\text{u}\text{?}\text{=}\text{f(u)}$ is usually a digital computer. Due to finiteness of the computer wordlength, the numerical solution $\text{u}{}_{\mathrm{c}}$ is in general different from u. Let $\text{u}\text{(}\text{x}\text{, t)}$ denote the actual response of the system in continuum at points corresponding to those of u. In the literature. $\text{u}\text{(}\text{x}\text{, t)-}\text{u}$ is called the discretization errors, $\text{u}\text{-}\text{u}{}_{\mathrm{c}}$ the round-off errors, and the s is. $\text{u}\text{(}\text{x}\text{, t)-}\text{u}{}_{\mathrm{c}}$ is called the solution errors. In this paper, a state-of-the-art survey is given on the identification, growth, relative magnitudes, estimation, and control of the components of the solution errors.     

Inhomogeneous Dirichlet boundary condition in the a posteriori error control of the obstacle problem

We study a posteriori error control of finite element approximation of the elliptic obstacle problem with nonhomogeneous Dirichlet boundary condition. The results in the article are two fold. Firstly, we address the influence of the inhomogeneous Dirichlet boundary condition in residual based a posteriori error control of the elliptic obstacle problem. Secondly by rewriting the obstacle problem in an equivalent form, we derive a posteriori error bounds which are in simpler form and efficient. To accomplish this, we construct and use a post-processed solution ${\stackrel{?}{u}}_h$ of the discrete solution $u_h$ which satisfies the exact boundary conditions sharply although the discrete solution $u_h$ may not satisfy. We propose two post processing methods and analyze them, namely the harmonic extension and a linear extension. The theoretical results are illustrated by the numerical results.     

A priori and a posteriori error analysis for a large-scale ocean circulation finite element model

We consider the finite element solution of the stream function–vorticity formulation for a large-scale ocean circulation model. First, we study existence and uniqueness of solution for the continuous and discrete problems. Under appropriate regularity assumptions we prove that the stream function can be computed with an error of order h in H¹-seminorm. Second, we introduce and analyze an h-adaptive mesh refinement strategy to reduce the spurious oscillations and poor resolution which arise when convective terms are dominant. We propose an a posteriori anisotropic error indicator based on the recovery of the Hessian from the finite element solution, which allows us to obtain well adapted meshes. The numerical experiments show an optimal order of convergence of the adaptive scheme. Furthermore, this strategy is efficient to eliminate the oscillations around the boundary layer.     

Adaptive Finite Element Approximation for an Elliptic Optimal Control Problem with Both Pointwise and Integral Control Constraints

In this paper, we study the adaptive finite element approximation for a constrained optimal control problem with both pointwise and integral control constraints. We first obtain the explicit solutions for the variational inequalities both in the continuous and discrete cases. Then a priori error estimates are established, and furthermore equivalent a posteriori error estimators are derived for both the state and the control approximation, which can be used to guide the mesh refinement for an adaptive multi-mesh finite element scheme. The a posteriori error estimators are implemented and tested with promising numerical results.     

On some convergence results for FDM with irregular mesh

The proof of convergence of the finite difference method with arbitrary irregular meshes for some class of elliptic problems is presented. By the use of the truncation error technique and stability analysis it was showed that $\text{max}{}_{\mathrm{i}}\text{¦u}{}^{\mathrm{i}}\text{? u}{}_{\mathrm{h}}{}^{\mathrm{i}}\text{¦? Ch}$, i.e., the solution u_h converges linearly with the size of the star. Correctness of this theorem was also confirmed by numerical tests.     

Convergence of a decoupled mixed FEM for the dynamic Ginzburg–Landau equations in nonsmooth domains with incompatible initial data

In this paper, we propose a fully discrete mixed finite element method for solving the time-dependent Ginzburg–Landau equations, and prove the convergence of the finite element solutions in general curved polyhedra, possibly nonconvex and multi-connected, without assumptions on the regularity of the solution. Global existence and uniqueness of weak solutions for the PDE problem are also obtained in the meantime. A decoupled time-stepping scheme is introduced, which guarantees that the discrete solution has bounded discrete energy, and the finite element spaces are chosen to be compatible with the nonlinear structure of the equations. Based on the boundedness of the discrete energy, we prove the convergence of the finite element solutions by utilizing a uniform \(L^{3+\delta }\) regularity of the discrete harmonic vector fields, establishing a discrete Sobolev embedding inequality for the Nédélec finite element space, and introducing a \(\ell ^2(W^{1,3+\delta })\) estimate for fully discrete solutions of parabolic equations. The numerical example shows that the constructed mixed finite element solution converges to the true solution of the PDE problem in a nonsmooth and multi-connected domain, while the standard Galerkin finite element solution does not converge.     

A stabilized finite volume method for the evolutionary Stokes–Darcy system

In this paper, we propose a stabilized fully discrete finite volume method based on two local Gauss integrals for a non-stationary Stokes–Darcy problem. This stabilized method is free of stabilized parameters and uses the lowest equal-order finite element triples $P1\text{–}P1\text{–}P1$ for approximating the velocity, pressure and hydraulic head of the Stokes–Darcy model. Under a modest time step restriction in relation to physical parameters, we give the stability analysis and the error estimates for the stabilized finite volume scheme by means of a relationship between finite volume and finite element approximations with the lower order elements. Finally, a series of numerical experiments are provided to demonstrate the validity of the theoretical results.     

A new a posteriori error estimator in adaptive direct boundary element methods: the Dirichlet problem

In this paper we propose a new a posteriori error estimator for a boundary element solution related to a Dirichlet problem with a second order elliptic partial differential operator. The method is based on an approximate solution of a boundary integral equation of the second kind by a Neumann series to estimate the error of a previously computed boundary element solution. For this one may use an arbitrary boundary element method, for example, a Galerkin, collocation or qualocation scheme, to solve an appropriate boundary integral equation. Due to the approximate solution of the error equation the proposed estimator provides high accuracy. A numerical example supports the theoretical results. Received: June 1999 / Accepted: September 1999     

Finite elements and characteristics applied to advection-diffusion equations

This paper deals with an algorithm for the solution of advection-diffusion equations based on the finite element method combined with the discretization of the total differential $\text{Dρ}\text{Dt}$. We give in the one dimensional case the finite difference analog of our Galerkin method. Diffusion of the scheme is studied in the two dimensional case by means of a classic example. We show that the scheme is stable, has no phase error and leads to simple problems at each time step.     

Fully discrete finite element method based on pressure stabilization for the transient Stokes equations

In this work, a new fully discrete stabilized finite element method is studied for the two-dimensional transient Stokes equations. This method is to use the difference between a consistent mass matrix and underintegrated mass matrix as the complement for the pressure. The spatial discretization is based on the P₁–P₁ triangular element for the approximation of the velocity and pressure, the time discretization is based on the Euler semi-implicit scheme. Some error estimates for the numerical solutions of fully discrete stabilized finite element method are derived. Finally, we provide some numerical experiments, compared with other methods, we can see that this novel stabilized method has better stability and accuracy results for the unsteady Stokes problem.     

Local Discontinuous Galerkin Finite Element Method and Error Estimates for One Class of Sobolev Equation

In this paper we present a numerical scheme based on the local discontinuous Galerkin (LDG) finite element method for one class of Sobolev equations, for example, generalized equal width Burgers equation. The proposed scheme will be proved to have good numerical stability and high order accuracy for arbitrary nonlinear convection flux, when time variable is continuous. Also an optimal error estimate is obtained for the fully discrete scheme, when time is discreted by the second order explicit total variation diminishing (TVD) Runge-Kutta time-marching. Finally some numerical results are given to verify our analysis for the scheme.     

Reliable and efficient a posteriori error estimates for finite element approximations of the parabolic p-Laplacian

We generalize the a posteriori techniques for the linear heat equation in Verfürth (Calcolo 40(3):195–212, 2003) to the case of the nonlinear parabolic $p$ -Laplace problem thereby proving reliable and efficient a posteriori error estimates for a fully discrete implicite Euler Galerkin finite element scheme. The error is analyzed using the so-called quasi-norm and a related dual error expression. This leads to equivalence of the error and the residual, which is the key property for proving the error bounds.     

An accurate algorithm for computing the eigenvalues of a polygonal membrane

For a polygonal domain Ω, we consider the eigenvalue problem Δu + λu = 0 in Ω, u = 0 on the boundary of Ω. Ω is decomposed into subdomains Ω₁, Ω₂,...; on each $\text{Ω}{}_{\mathrm{i}}$, u is approximated by a linear combination of functions which satisfy the equation Δu + Δu = 0 and continuity conditions are imposed at the boundaries of the subdomains. We propose a non-conventional method based on the use of a Rayleigh quotient. We present numerical examples and a proof of the exponential convergence of the algorithm.     

Curved finite element methods for the solution of singular integral equations on surfaces in R3

We have shown in [1]that the singular integral equation (1.2) on a closed surface Γ of R³ admits a unique solution q and is variational and coercive in the Hilbert space $\text{H}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$(Γ). In this paper, with the help of curved finite elements, we introduce an approximate surface Γ_h, and an approximate problem on Γ_h, whose solution is q_h. Then we study the error of approximation |q ? q_h| in some Hubert spaces and also the associated error |u ? u_h| of the potential.     

On error estimates of the fully discrete penalty method for the viscoelastic flow problem

In this paper, a fully discrete finite element penalty method is presented for the two-dimensional viscoelastic flow problem arising in the Oldroyd model, in which the spatial discretization is based on the finite element approximation and the time discretization is based on the backward Euler scheme. Moreover, we provide the optimal error estimate for the numerical solution under some realistic assumptions. Finally, some numerical experiments are shown to illustrate the efficiency of the penalty method.     

Some numerical investigations in nonlinear optics

The problem of the self-focusing of a light beam in nonlinear media is the central problem in nonlinear optics. The powerful laser beam propagation through a real medium under certain conditions is accompanied with such a phenomenon.Mathematically the problem deals with the investigation of the asymptotic behaviour of the solution of the parabolic equation

$\text{2i}\text{?u}\text{?z}\text{=}\text{?}{}^2\text{u}\text{?r}{}^2\text{+}\text{1}\text{r}\text{?u}\text{?r}\text{+?(∣u∣}{}^2\text{)u}$

with given initial distribution u(r,0) and boundary condition u(∞, z) = 0 where u is the electromagnetic field amplitude, f is a function which describes the refractive index deviation from its constant value in the linear medium. It is complex in the case of nonconservative media. In our investigation we combine analytical and numerical methods. The computational study of the self-focusing problem is complicated due to the boundary condition at infinity and the abrupt light amplitude behaviour in the paraxial region. We managed to overcome these difficulties by introducing the socalled quasi-uniform grid for the radial variable and by using the special technique of the correct transfer of the boundary condition from infinity. The main physical results are: (1) the conditions for the light self-trapping and waveguide creation are found, (2) the self-focusing mechanism and the law of increasing beam amplitude when approaching the collapse point are discovered; (3) the influence of the different kinds of absorption is investigated and the process of light “turbulence” is explained;All the analytical and numerical results are comparable with the experimental situation as well as with treatments by other authors.